Corollary

— Let F be a metrizable uniform space.

(i) Every sequence $(f_n)$ of elements of $\mathscr{S}(A, \mu; F)$ that converges locally almost everywhere to a mapping $f$ (necessarily $\mu$-measurable) of A into F, converges in measure to $f$ in A.

(ii) Let $(f_n)$ be a sequence of elements of $\mathscr{S}(A, \mu; F)$ that converges in measure to a mapping $f$ of A into F. For every set $B \subset A$ that is a countable union of integrable sets, there exists a subsequence $(f_{n_k})$ of $(f_n)$ such that the sequence $(f_{n_k}(x))$ converges in F to $f(x)$ for almost every $x \in B$.

(i) The assertion follows at once from the extension of Egoroff’s theorem mentioned in No. 10.

(ii) By Lemma 4, there exists a subsequence $(f_{n_k})$ of $(f_n)$ such that $(f_{n_k}(x))$ is a Cauchy sequence in F for every $x \in B - N$, where N is negligible; let $f'(x) \in \widehat{F}$ be the limit of this sequence for $x \in B - N$. It is clear that $f'$ is a $\mu$-measurable mapping of $B - N$ into $\widehat{F}$, and the sequence $(f_n)$ converges in measure to $f'$ in $B - N$ by (i); $f'$ is therefore equal to $f$ almost everywhere in B.